A123692 -id:A123692 - cp's OEIS frontend

A128669 Primes p such that p^2 divides 23^(p-1) - 1.

Original entry on oeis.org

13, 2481757, 13703077, 15546404183, 2549536629329

Offset: 1

Alexander Adamchuk, Mar 26 2007

No further terms up to 3.127*10^13.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 233.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2)

Cf. A001220, A014127, A123692, A123693, A128667, A128668, A090968, A039951

Select[Prime[Range[5*10^7]], Mod[ 23^(# - 1) - 1, #^2] == 0 &] (* G. C. Greubel, Jan 18 2018 *)
Select[Prime[Range[93*10^9]],PowerMod[23,#-1,#^2]==1&] (* Harvey P. Dale, May 15 2018 *)

A126197 GCDs arising in A126196.

Original entry on oeis.org

11, 1093, 1093, 3511, 3511, 5557, 104891, 1006003

Offset: 1

Max Alekseyev and Tanya Khovanova, Mar 07 2007

All terms are primes. Note a connection to the Wieferich primes A001220: a(2) = a(3) = A001220(1), a(3) = a(4) = A001220(2).
From John Blythe Dobson, Jan 14 2017: (Start)
All Wieferich primes p will belong to this sequence twice, because if H([p/k]) denotes the harmonic number with index floor(p/k), then p divides all of H([p/4]), H([p/2]), and H(p-1). The first two of these elements gives one solution, and the second and third another. This property of the Wieferich primes predates their name, and was apparently first proved by Glaisher in "On the residues of r^(p-1) to modulus p^2, p^3, etc.," pp. 21-22, 23 (see References).
Note also a connection to the Mirimanoff primes A014127: a(1) = A014127(1), a(8) = A014127(2). All Mirimanoff primes p will belong to this sequence, because p divides both H([p/3]) and H([2p/3]). This property of the Mirimanoff primes likewise predates their name, and was apparently first proved by Glaisher in "A general congruence theorem relating to the Bernoullian function," p. 50 (see Links).
The Wieferich primes and Mirimanoff primes would seem to be the only cases for which the value of n in A126196(n) is predictable from knowledge of p. It is not obvious that all members of the present sequence are prime; however, by definition all their divisors must be non-harmonic primes A092102. Furthermore, it is clear from the cited literature under that entry that H([n/2]) == H(n) == 0 (mod p) is only possible when n < p. Thus, all divisors of the present sequence must belong to the harmonic irregular primes A092194.
One possible reason for interest in this sequence is a 1995 result of Dilcher and Skula (see Links) which among other things shows that if a prime p were an exception to the first case of Fermat's Last Theorem, then p would divide both H([p/k]) and H([2p/k]) for every value of k from 2 to 46. To date, the only values for which such coincidences have been found have k = 2, 3, or 4. For k = 6 to hold, p would have to be simultaneously a Wieferich prime and a Mirimanoff prime, while for k = 5 to hold, p would have to be simultaneously a Wall-Sun-Sun prime and a member of A123692. The sparse numerical results for the present sequence suggest that even the more relaxed condition H([n/2]) == H(n) == 0 (mod p) is rarely satisfied. (End)

J. W. L. Glaisher, On the residues of r^(p-1) to modulus p^2, p^3, etc., Quarterly Journal of Pure and Applied Mathematics 32 (1900-1901), 1-27.

Karl Dilcher and Ladislav Skula, A new criterion for the first case of Fermat's Last Theorem, Mathematics of Computation 64 (1995), 363-392.
J. W. L. Glaisher, A general congruence theorem relating to the Bernoullian function, Proceedings of the London Mathematical Society 33 (1900-1901), 27-56.

Cf. A001008, A001220, A002805, A058313, A074791, A121594, A125581, A126196.

f[n_] := GCD @@ Numerator@ HarmonicNumber@ {n, Floor[n/2]}; f@ Select[ Range[5000], f[#] > 1 &] (* Giovanni Resta, May 13 2016 *)

a(8) from Giovanni Resta, May 13 2016

A212583 Primes p such that p^2 divides 6^(p-1) - 1.

Original entry on oeis.org

66161, 534851, 3152573

Offset: 1

Felix Fröhlich, May 22 2012

Base 6 Wieferich primes.

Next term > 4.119*10^13. [See Fischer link]

P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1996, page 347

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table p. 5.
François G. Dorais and Dominic Klyve, A Wieferich prime search up to p < 6.7*10^15, J. Integer Seq. 14 (2011), Art. 11.9.2, 1-14.
Richard Fischer, Thema: Fermatquotient B^P-1 == 1 mod (P^2)
Wilfrid Keller and Jörg Richstein, Fermat quotients q_p(a) that are divisible by p.
Eric Weisstein, Fermat Quotient, MathWorld
Wikipedia, Base-a Wieferich primes

Cf. A001220, A014127, A090968, A123692, A123693, A128667, A128668, A128669.

Select[Prime[Range[1000000]], PowerMod[6, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

N=10^9; default(primelimit,N);
forprime(n=2,N,if(Mod(6,n^2)^(n-1)==1,print1(n,", ")));
\\ Joerg Arndt, May 01 2013

A242959 Numbers n such that 5^A000010(n) == 1 (mod n^2).

Original entry on oeis.org

2, 20771, 40487, 41542, 80974, 83084, 161948, 643901, 1255097, 1287802, 1391657, 1931703, 2510194, 2575604, 2783314, 3765291, 3863406, 4174971, 5020388, 5151208, 5566628, 7530582, 7726812, 8349942, 10040776, 11133256, 15061164, 15308227, 15453624, 16699884

Offset: 1

Felix Fröhlich, May 27 2014

nonn

Giovanni Resta, Table of n, a(n) for n = 1..479 (terms < 10^15, first 75 terms from Felix Fröhlich)

Cf. A077816, A242958, A242960.

If a(n) is prime, then a(n) is in A123692.

Select[Range[167*10^5],PowerMod[5,EulerPhi[#],#^2]==1&] (* Harvey P. Dale, Jun 02 2020 *)

for(n=2, 10^9, if(Mod(5, n^2)^(eulerphi(n))==1, print1(n, ", ")))

A055578 "Non-generous primes": primes p whose least positive primitive root is not a primitive root of p^2.

Original entry on oeis.org

2, 40487, 6692367337

Offset: 1

Bernard Leak (bernard(AT)brenda-arkle.demon.co.uk), Aug 24 2000

For r a primitive root of a prime p, r + qp is a primitive root of p: but r + qp is also a primitive root of p^2, except for q in some unique residue class modulo p. In the exceptional case, r + qp has order p-1 modulo p^2 (Burton, section 8.3).
No other terms below 10^12 (Paszkiewicz, 2009).
Each term p is a Wieferich prime to base A046145(p). For example, a(2) and a(3) are base-5 Wieferich (see A123692). - Jeppe Stig Nielsen, Mar 06 2020

David Burton, Elementary Number Theory, Allyn and Bacon, Boston, 1976, first edition (cf. Section 8.3).

Joerg Arndt, Matters Computational (The Fxtbook), section 39.7.2, p.780.
Stephen Glasby, Three questions about the density of certain primes, Posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Apr 22, 2001.
Bryce Kerr, Kevin McGown, Tim Trudgian, The least primitive root modulo p^2, arXiv:1908.11497 [math.NT], 2019.
A. Paszkiewicz, A new prime for which the least primitive root (mod p) and the least primitive root (mod p^2) are not equal, Math. Comp. 78 (2009), 1193-1195.

Cf. A060503, A060504.

Select[Prime@Range[7!], ! PrimitiveRoot[#] == PrimitiveRoot[#^2] &] (* Arkadiusz Wesolowski, Sep 06 2012 *)

Prime A000040(n) is in this sequence iff A001918(n)^(A000040(n)-1) == 1 (mod A000040(n)^2).

Prime A000040(n) is in this sequence iff A001918(n) differs from A127807(n).

a(3) from Stephen Glasby (Stephen.Glasby(AT)cwu.EDU), Apr 22 2001

Edited by Max Alekseyev, Nov 10 2011

A242741 Primes p such that p^2 divides 15^(p-1) - 1.

Original entry on oeis.org

29131, 119327070011

Offset: 1

Felix Fröhlich, May 21 2014

Base 15 Wieferich primes. According to Richard Fischer there is no other term up to approximately 5*10^13.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
R. Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)

Cf. A001220, A014127, A123692, A212583, A123693, A045616, A111027, A128667, A234810

Select[Prime[Range[1000000]], PowerMod[15, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

forprime(n=2, 10^9, if(Mod(15, n^2)^(n-1)==1, print1(n, ", ")));

A242982 Primes p such that p^2 divides 20^(p-1) - 1.

Original entry on oeis.org

281, 46457, 9377747, 122959073

Offset: 1

Felix Fröhlich, May 28 2014

Base 20 Wieferich primes. According to Richard Fischer, there is no other term up to approximately 5*10^13.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
R. Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)

Cf. A001220, A014127, A123692, A212583, A123693, A045616, A111027, A039951, A096082.

Select[Prime[Range[1000000]], PowerMod[20, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

forprime(n=2, 10^9, if(Mod(20, n^2)^(n-1)==1, print1(n, ", ")));

A244260 Primes p such that p^2 divides 18^(p-1) - 1.

Original entry on oeis.org

5, 7, 37, 331, 33923, 1284043

Offset: 1

Felix Fröhlich, Jun 24 2014

Base 18 Wieferich primes. According to Richard Fischer there is no other term up to approximately 5*10^13.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
R. Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)

Cf. A001220, A014127, A123692, A212583, A123693, A045616, A111027, A128667, A234810, A242741, A128668, A090968, A242982, A039951, A096082.

Select[Prime[Range[1000000]], PowerMod[18, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

forprime(n=2, 10^9, if(Mod(18, n^2)^(n-1)==1, print1(n, ", ")));

A125775 Numbers k such that 5^k mod k = 5^k mod k^2.

Original entry on oeis.org

1, 2, 4, 5, 6, 12, 25, 42, 52, 84, 125, 156, 186, 372, 625, 1092, 1218, 1302, 1806, 2436, 2604, 2756, 3125, 3612, 4836, 5334, 7212, 8268, 10668, 12324, 15625, 15918, 18858, 19140, 20771, 24492, 26080, 31668, 31836, 33852, 37716, 37758, 40487, 41542

Offset: 1

Alexander Adamchuk, Dec 07 2006

nonn

Includes all powers of 5 (A000351).

a(2) = 2, a(4) = 5, a(35) = 20771 and a(43) = 40487 are the only listed primes. More known primes are listed in A123692.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000351, A123692, A068535, A125773, A125774.

Do[f=PowerMod[5,n,n];g=PowerMod[5,n,n^2];If[f==g,Print[n]],{n,1,1000000}]
Select[Range[42000],PowerMod[5,#,#]==PowerMod[5,#,#^2]&] (* Harvey P. Dale, Aug 20 2022 *)

A126432 Double Wieferich primes with q = 5.

Original entry on oeis.org

1645333507, 188748146801

Offset: 1

Alexander Adamchuk, Mar 12 2007

Double Wieferich prime pairs are pairs of primes (p, q) such that q^(p-1) == 1 (mod p^2) and p^(q-1) == 1 (mod q^2).

C. K. Caldwell, The Prime Glossary, Fermat quotient.
Wikipedia, Wieferich pair

Subsequence of A123692.

Cf. A124121, A124122, A125775.

cp's OEIS Frontend

A128669 Primes p such that p^2 divides 23^(p-1) - 1.

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A126197 GCDs arising in A126196.

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A212583 Primes p such that p^2 divides 6^(p-1) - 1.

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A242959 Numbers n such that 5^A000010(n) == 1 (mod n^2).

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A055578 "Non-generous primes": primes p whose least positive primitive root is not a primitive root of p^2.

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A242741 Primes p such that p^2 divides 15^(p-1) - 1.

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A242982 Primes p such that p^2 divides 20^(p-1) - 1.

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A244260 Primes p such that p^2 divides 18^(p-1) - 1.

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